IIT JEE Main Maths -Unit 6- Relation between A.M. G.M,H.M..- Study Notes-New Syllabus

Harmonic Progression (H.P.) and Relation Between A.P., G.P., and H.P.

A Harmonic Progression (H.P.) is a sequence of numbers such that the reciprocals of its terms form an Arithmetic Progression (A.P.).

$ a_1, a_2, a_3, \dots \text{ are in H.P. if } \dfrac{1}{a_1}, \dfrac{1}{a_2}, \dfrac{1}{a_3}, \dots \text{ are in A.P.}$

General Form of an H.P.

If \( \dfrac{1}{a}, \dfrac{1}{a + d}, \dfrac{1}{a + 2d}, \dots \) are reciprocals forming an A.P., then:

$\text{H.P.} = \dfrac{1}{a},\, \dfrac{1}{a + d},\, \dfrac{1}{a + 2d},\, \dots $

nth Term of an H.P.

If \( \dfrac{1}{a_1}, \dfrac{1}{a_2}, \dots \) are in A.P., then the \( n \)th term of the H.P. is:

$a_n = \dfrac{1}{A + (n – 1)d} $

where \( A = \dfrac{1}{a_1} \) and \( d \) = common difference of the A.P. formed by reciprocals.

Harmonic Mean (H.M.)

The Harmonic Mean between two numbers \( a \) and \( b \) is given by:

$ H = \dfrac{2ab}{a + b}$

If three numbers \( a, H, b \) are in H.P., then their reciprocals \( \dfrac{1}{a}, \dfrac{1}{H}, \dfrac{1}{b} \) are in A.P.

Inserting Harmonic Means

If \( n \) harmonic means are inserted between \( a \) and \( b \), then their reciprocals form an A.P. between \( \dfrac{1}{a} \) and \( \dfrac{1}{b} \).

Hence, the reciprocals of the inserted H.M.s are:

$\dfrac{1}{a} + k\left(\dfrac{1}{b} – \dfrac{1}{a}\right)\dfrac{1}{n + 1}, \quad k = 1, 2, 3, \dots, n $

and their H.M.s are the reciprocals of these values.

Relationship Between A.P., G.P., and H.P.

For any two positive numbers \( a \) and \( b \):

$ A.M. \ge G.M. \ge H.M. $

and equality holds only when \( a = b \).

Also,

$ A.M. \times H.M. = (G.M.)^2 $

Relationship Between Sequences

• If \( a, b, c \) are in A.P., then \( \dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c} \) are in H.P.
• If \( a, b, c \) are in G.P., then \( \log a, \log b, \log c \) are in A.P.
• Conversely, if \( \dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c} \) are in A.P., then \( a, b, c \) are in H.P.

 Key Takeaways for JEE:

• H.P. is the reciprocal of A.P.
• If \( a, b, c \) are in A.P. → \( \dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c} \) are in H.P.
• \( H = \dfrac{2ab}{a + b} \) and \( A \times H = G^2 \).
• When inserting H.M.s, work with reciprocals — they form an A.P.
• \( A \ge G \ge H \) is a fundamental inequality used in JEE inequality and optimization questions.

 Combined Comparison Table (A.P., G.P., and H.P.)